OFFPRINT
A unified framework for force-based and energy-based adaptive resolution simulations Karsten Kreis, Davide Donadio, Kurt Kremer and Raffaello Potestio EPL, 108 (2014) 30007
Please visit the website www.epljournal.org Note that the author(s) has the following rights: – immediately after publication, to use all or part of the article without revision or modification, including the EPLAformatted version, for personal compilations and use only; – no sooner than 12 months from the date of first publication, to include the accepted manuscript (all or part), but not the EPLA-formatted version, on institute repositories or third-party websites provided a link to the online EPL abstract or EPL homepage is included. For complete copyright details see: https://authors.epletters.net/documents/copyright.pdf.
A LETTERS JOURNAL EXPLORING THE FRONTIERS OF PHYSICS
AN INVITATION TO SUBMIT YOUR WORK www.epljournal.org
The Editorial Board invites you to submit your letters to EPL EPL is a leading international journal publishing original, innovative Letters in all areas of physics, ranging from condensed matter topics and interdisciplinary research to astrophysics, geophysics, plasma and fusion sciences, including those with application potential. The high profile of the journal combined with the excellent scientific quality of the articles ensures that EPL is an essential resource for its worldwide audience. EPL offers authors global visibility and a great opportunity to share their work with others across the whole of the physics community.
Run by active scientists, for scientists EPL is reviewed by scientists for scientists, to serve and support the international scientific community. The Editorial Board is a team of active research scientists with an expert understanding of the needs of both authors and researchers. www.epl journal.org
ISSN 0295-5075
www.epl journal.org
ISSN 0295-5075
www.epl journal.org
ISSN 0295-5075
A L ETTERS J OURNAL E XPLORING THE F RONTIERS OF P HYSICS
A L ETTERS J OURNAL E XPLORING THE F RONTIERS OF P HYSICS
A L ETTERS J OURNAL E XPLORING THE F RONTIERS OF P HYSICS
Volume 105 Number 1 January 2014
Volume 104 Number 1 October 2013
Volume 103 Number 1 July 2013
www.epljournal.org
A LETTERS JOURNAL EXPLORING THE FRONTIERS OF PHYSICS
OVER
560,000
www.epljournal.org
Six good reasons to publish with EPL We want to work with you to gain recognition for your research through worldwide visibility and high citations. As an EPL author, you will benefit from:
full text downloads in 2013
1
Quality – The 50+ Co-editors, who are experts in their field, oversee the entire peer-review process, from selection of the referees to making all final acceptance decisions.
average accept to online publication in 2013
2
Convenience – Easy to access compilations of recent articles in specific narrow fields available on the website.
10,755
3
Speed of processing – We aim to provide you with a quick and efficient service; the median time from submission to online publication is under 100 days.
4
High visibility – Strong promotion and visibility through material available at over 300 events annually, distributed via e-mail, and targeted mailshot newsletters.
5
International reach – Over 2600 institutions have access to EPL, enabling your work to be read by your peers in 90 countries.
6
Open access – Articles are offered open access for a one-off author payment; green open access on all others with a 12-month embargo.
24 DAYS
citations in 2013
“We greatly appreciate the efficient, professional and rapid processing of our paper by your team.” Cong Lin Shanghai University
Details on preparing, submitting and tracking the progress of your manuscript from submission to acceptance are available on the EPL submission website www.epletters.net. If you would like further information about our author service or EPL in general, please visit www.epljournal.org or e-mail us at
[email protected]. EPL is published in partnership with:
European Physical Society
Società Italiana di Fisica
www.epljournal.org
EDP Sciences
IOP Publishing
November 2014 EPL, 108 (2014) 30007 doi: 10.1209/0295-5075/108/30007
www.epljournal.org
A unified framework for force-based and energy-based adaptive resolution simulations Karsten Kreis1,2 , Davide Donadio1 , Kurt Kremer1 and Raffaello Potestio1 1 2
Max Planck Institute for Polymer Research - Ackermannweg 10, 55128 Mainz, Germany Graduate School Materials Science in Mainz - Staudinger Weg 9, 55128 Mainz, Germany received 15 August 2014; accepted in final form 20 October 2014 published online 10 November 2014 PACS PACS PACS
05.10.-a – Computational methods in statistical physics and nonlinear dynamics 02.70.Ns – Molecular dynamics and particle methods 82.20.Wt – Computational modeling; simulation
Abstract – Adaptive resolution schemes enable molecular dynamics simulations of liquids and soft matter employing two different resolution levels concurrently in the same setup. These methods are based on a position-dependent interpolation of either forces or potential energy functions. While force-based methods generally lead to non-conservative forces, energy-based ones include undesired force terms proportional to the gradient of the interpolation function. In this work we establish a so far missing bridge between these formalisms making use of the generalized Langevin equation, thereby providing a unifying framework to traditionally juxtaposed approaches to adaptive simulations. c EPLA, 2014 Copyright
Simulation methods in which two different descriptions of the same system (all atom/coarse grained, quantum mechanical/classical . . .) are concurrently employed have been developed and applied since the early times of computational science [1–34]. In these schemes only a small region of the simulation domain is treated with an accurate, computationally expensive model (e.g. an atomistic model), while for the rest a simpler and more efficient representation (e.g. coarse-grained) is employed. In most methods the description of a molecule is fixed for the whole duration of the simulation. Approaches in which the molecules resolution changes “on the fly” according to its position in space, e.g. due to diffusion, are dubbed adaptive resolution methods. In the present work we focus on the latter, which enable the simulation of soft matter, and specifically of liquids. Furthermore, we concentrate only on classical systems, and do not address quantum mechanics-molecular mechanics models. Concerning particle-based, classical simulations, two main classes of adaptive resolution simulation schemes can be identified: force-based and energy-based. The first ones [25–27] share the common feature that the two models employed are merged via a position-dependent interpolation of the force acting between molecule pairs; in the second case [31–34] this interpolation is performed at the level of potentials. The direct interpolation of forces enables the instantaneous preservation of Newton’s third
law [25], often desired in molecular dynamics simulations to correctly describe hydrodynamics [35]. This comes at the price of having an intrinsically non-conservative forcefield which requires a local thermostat to enforce thermodynamic equilibrium and stability [36,37]. Energy-based schemes are conservative and Hamiltonian by construction, allowing an explicit, partition function-based theoretical treatment as well as energy-conserving MD simulations or Monte Carlo simulations [33,34]. On the other hand, the forces obtained differentiating these energy functions contain terms proportional to the gradient of the position-dependent function employed to interpolate between the models. Such terms cannot be written as a sum of pairwise antisymmetric terms; therefore, they break Newton’s third law and momentum conservation, and are generally seen as an undesired complication. However, they are only present where the gradient of the position-dependent interpolation function does not disappear, i.e. only in the interpolation region, hence leaving Newton’s third law in the bulk high- and low-resolution regions valid. Removing these terms would effectively transform an energy-based approach into a force-based one. Exact momentum conservation and energy conservation cannot be simultaneously achieved in an adaptive resolution simulation [37]. This fundamental limitation has hampered the widespread application of adaptive
30007-p1
Karsten Kreis et al. resolution methods in soft matter. Advantages and limitations of force-based and energy-based dual-resolution schemes have been largely investigated [26,33,34,36–38]. Nonetheless, it is still necessary to set a unified framework that would encompass the two classes of methods and contextualize connections and differences between them. In the present work we pursue this goal by establishing a formal relation between an energy-based method, the Hamiltonian adaptive resolution simulation scheme (H-AdResS [33,34]), and a force-based scheme derived from it. We begin by considering two Hamiltonians, both having the same number of molecules: Vαr , r = 0, 1, (1) Hr = K +
effect can be compensated by adding a conservative external field, acting only in the hybrid region, i.e. in the region where 0 < λ < 1. This methodology, aimed at preserving the appropriate thermodynamic conditions in each subdomain, has been applied in both force-based [26] and energy-based [33,34] approaches. A further source of thermodynamic imbalance is given by the last force of eq. (3), dubbed drift force. This term is specific to the energy-based schemes and represents the main difference with respect to force-based approaches to adaptive resolution simulations. Therefore, it is on this quantity that we now focus our attention. Specifically, we define two different models, named E and F , with the following equations of motions: dr ¨ α = FN model E : mα x α + Fα , ¨ α = FN model F : mα x α.
α
where K = α p2α /2mα is the total kinetic energy. Without loss of generality we assume the molecules to be pointlike particles; the extension to multi-atomic molecules is straightforward. The single-molecule potentials Vαr are the sums of all intermolecular potentials acting on molecule α, properly normalized so that double counting is avoided [33,34]. In most of the applications [30,39] a fully atomistic (r = 1) and a coarse-grained (r = 0) potential are used. In the following, though, no assumption is made on the form of these interactions other than finite range. These Hamiltonians are interpolated, in the H-AdResS scheme [33], with a resolution function λα = λ(Rα ), varying monotonically between 1 and 0 as a function of the position of the molecule Rα . This Hamiltonian Hmix and the force Fα acting on molecule α are Hmix = K + (2) λα Vα1 + (1 − λα )Vα0 ,
In the first case the equations of motion are integrated including all the contributions obtained differentiating the mixed Hamiltonian Hmix ; in the second case the drift force is removed, e.g. as it is done in [27], and the resulting model is a representative of the class of force-based approaches to dual-resolution simulations. Our goal is now to understand what is the difference between these two models, and specifically under what conditions one can be considered approximately close or equivalent to the other. As a first step we remove the average of the drift force from the total force of eq. (6) acting on molecule α. This can be done, as demonstrated in refs. [33,34], by adding to the mixed Hamiltonian an external field f (λ) depending only on the resolution of the molecule:
mix = Hmix − H
α
dr Fα = FN α + Fα ,
(3)
with FN α =
λα + λβ β
2
F1α|β
1 0 Fdr α = −[Vα − Vα ]∇λα ,
λα + λβ 0 + 1− Fα|β , 2
(4) (5)
where we introduced the total force Frα|β acting on molecule α due to the interaction V r with molecule β. Without loss of generality for our following derivation, we have explicitly written the forces for the case of pairwise interactions [33,34]. The term FN α contains all the derivatives of the potential energy functions Vαr with respect to the position of molecule α; the superscript N indicates that if the V 0 and V 1 potentials separately satisfy Newton’s third law, then so does the force FN α [33]. The second term, the drift force Fdr α , is the contribution coming from differentiating the resolution function. We note here that, in general, the two models have different virial pressure for the same state point, which in turn determines a non-uniform density in the system. This
(6) (7)
N
f (λα ),
(8)
α=1
dr ′
dr F α = Fα + f (λα )∇λα .
(9)
Employing for f (λ) the Helmholtz free energy per particle as computed in a Kirkwood thermodynamic integration [40], the drift force is on average removed,
dr i.e. F α = 0. This Free Energy Compensation (FEC) method demonstrates [33,34] that, in the H-AdResS framework, a seamless coupling between two arbitrary models of the same system is obtained by removing the difference in chemical potential. This result not only provides a deeper understanding of the fundamental physics of these approaches, but also enables us to operate with them in a computationally efficient manner.
dr averages to zero by construction; thereThe force F α fore, the conservative force field originating from the Hamiltonian in eq. (8) is equal to that of model F , with the exception of a fluctuating term. The absence of the latter from model F causes its force-field to be non-conservative. It has been empirically observed [33,36,38] that if a symplectic, i.e. energy and phase space volume-conserving integrator is employed to evolve the equations of motion of
30007-p2
A unified framework for force-based and energy-based adaptive resolution simulations a force-based dual-resolution scheme, the kinetic energy of the system steadily increases. To remove this “excess heat” produced by molecules crossing the hybrid region, a local thermostat, such as Langevin’s, is required. Provided that the coupling is sufficiently strong, this procedure maintains the system in a steady state and avoids energy drift [25,26]. In order to better understand the role played by the fluctuations of the drift force we performed a microcanonical H-AdResS simulation. We chose as a test system a liquid of 1596 molecules in a box of dimensions 73.69σ × 15σ×15σ. The molecules are composed of four atoms connected by anharmonic bonds. In the low-resolution region each molecule is described as a single sphere. The interactions between atoms of different molecules is provided by a purely repulsive Weeks Chandler Andersen (WCA) potential [41]: ⎧ σ 12 σ 6 1 ⎨ 4ǫ − + , if r ≤ 21/6 σ, (10) V1 = r r 4 ⎩ 0, if r > 21/6 σ,
This system is the same as in ref. [33], except for the box size along the X-direction, which has been doubled; the simulations have been performed with the ESPResSo++ [42] package. To maintain the same density in both regions and remove, on average, the drift force, a FEC has been applied. The equations of motion for a molecule in this setup read dr
¨ α = FN mα x α + Fα .
(14)
The second step is to compute the autocorrelation func dr . This quantion of the average-subtracted drift force F tity, reported in fig. 1 (top panel, thicker line), provides interesting information about the statistical properties of the drift force. Specifically, we observe that its normalized autocorrelation function decays with a finite characteristic time, which we estimate ≃ 0.15τ by a simple exponential fit. This autocorrelation is obtained averaging over various particles in different parts of the hybrid region. To obtain a more detailed description of the dynamics of the system we compute the autocorrelation function of the drift force where r = |rα,i − rβ,j |. The coarse-grained molecules for a specific value of λ. To this end, we performed 10 interact via a hard-sphere potential: simulations of our system employing a mixed-resolution 12 Hamiltonian in which the resolution λ is the same for all 1.7σ V 0 = 4ǫ . (11) molecules: r − 0.05σ (15) Hλ = K + λU 1 + (1 − λ)U 0 , where U r = α Vαr and λ = 0.1n + 0.05, n = 0, 1, · · · 9. From these simulations we computed, for a given molecule, ΔV (t) = V 1 (t)−V 0 (t)−V 1 −V 0 as a function of time at different λ values; the autocorrelation of these quantities is then measured and related to the force autocorrelation (12) by
Given these potential energy functions, we define the characteristic time scale of the system as τ = σ m/ǫ, where the mass is m = 1. In the following, the quantities dimensioned as time will be expressed in units of τ . Finally, the intra-molecular interaction is given by a quartic bond potential: V int (rint ) =
1 2 k(rint − b2 )2 4
with rint = |rα,i − rα,j |2 being the distance between atoms within the same molecule α. Furthermore, we chose b = σ and k = 7500 σǫ2 . In the simulations, the molecules change their resolution along the X direction with the center of the atomistic, high-resolution region being fixed at the center of the box. The total size of the atomistic region is dat = 20σ and the thickness of the hybrid region is dhy = 10σ; the rest of the system is treated with the coarse-grained model. To assign to a molecule its positiondependent resolution λ, its distance from the boundary between the atomistic and the hybrid region is computed, i.e. |Rx | − dat /2, where Rx denotes the X coordinate of the molecule’s center of mass in a coordinate system with origin at the center of the simulation box. This quantity is then inserted into the resolution function λ(x), which is given as ⎧ 1 : x ≤ 0, ⎪ ⎪ ⎪ ⎨ 2 d 30 1 5 dhy 4 hy 3 λ(x) = 1− 5 x − x + x : 0 < x < dhy , ⎪ d 5 2 3 ⎪ hy ⎪ ⎩ 0 : x ≥ dhy . (13)
dr
dr
(0) = (∇λ)2 ΔV (t)ΔV (0),
(t)F F
(16)
where we made use of (see also eq. (5) and eq. (9))
dr (t) = −∇λ V 1 (t) − V 0 (t) − V 1 − V 0 . F
(17)
Also the autocorrelations so measured are reported in fig. 1 (top panel, thinner lines): their decay times increase with increasing value of the resolution. The origin of this behavior lies in the longer time required to molecules interacting via the atomistic potential V 1 to rearrange their internal structure; this, in turn, determines slower changes in the drift force acting on a molecule with high resolution. All the autocorrelations, though, decay to negligible values within 1 reduced time unit. It is instructive to measure the change of resolution that a molecule in the hybrid region undergoes by diffusing. In fig. 1 (bottom panel) we show the average resolution change Δλ = |λ(x − δx/2) − λ(x + δx/2)| of a molecule √ which moves from the position x by a distance δx = 2Dλ τλ ; the latter quantity makes use of the diffusion constant Dλ and the drift force autocorrelation decay
30007-p3
Karsten Kreis et al.
dr is a zero-average fluctuating force with drift force F a finite autocorrelation time: we can therefore apply to the system a colored noise with memory kλα such that it instantaneously and exactly cancels the drift force, i.e.
dr . This model has the following equations of kλα = −F α motion: dr
λ N
¨ α = FN mα x α + Fα + kα = Fα .
Fig. 1: (Color online) Top: autocorrelation of the drift force computed for different λ’s. The arrow on the plot indicates the direction of increasing λ. The thicker black line is the autocorrelation of the drift force of a particle in the hybrid region. Inset: autocorrelation of the drift force at t = 0, as a function of λ. Bottom: average change of resolution due to a molecule’s displacement during the characteristic decay time of the corresponding drift force autocorrelation.
time τλ as measured in the aforementioned simulations performed at uniform resolution. What we observe is that in the time it takes to the autocorrelation to decay to negligible values, the resolution of the molecules changes at most by 4%, a change which occurs mainly in the center of the hybrid region where the gradient of λ is particularly steep. The results in fig. 1 imply that the drift force decorrelates before a molecule substantially changes its resolution, i.e. the λ parameter, by diffusing. This observation is not expected a priori to hold for every system; however, the necessary conditions to obtain this behavior can always be achieved by appropriately choosing the size of the hybrid region as a function of the diffusivity. In general, a system-specific choice of the setup parameters enables the successful simulation of more complicated systems than our test model [39,43–45].
(18)
In eq. (18) the colored noise is explicitly dependent on the instantaneous resolution of the molecule, as indicated by the superscript λ. The resulting system, rightmost term in eq. (18), is a force-based model F . This interpretation manifestly explains the previously mentioned fact that a force-based dual-resolution simulation scheme undergoes a systematic kinetic energy increase when integrated without a local thermostat [33,36,38]: the absence of the drift force can be equated with the introduction of an autocorrelated random noise in a Hamiltonian, energyconserving scheme. To enforce thermal and mechanical equilibrium, the autocorrelated random noise must be balanced by a history-dependent friction term according to the fluctuation-dissipation theorem. The application of the colored noise and the friction with memory amounts to thermostat a model E by means of a Generalized Langevin Equation (GLE) [46–51]:
dr ¨ α = FN mα x α + Fα t − −∞ Kλ (t − t′ )vα (t′ )dt′ + kλα , dr
(19)
dr
(t)F
(t′ )λ , Kλ (t − t′ ) = βF
where the subscript in the average indicates that only those molecules whose resolution is λ are considered. Our definition of the friction kernel relies on the assumption that the molecule’s resolution can be considered approximately constant throughout the integration time when Kλ = 0; such assumption is justified by the previous observation that the force autocorrelation time decays to negligible values before the molecule can substantially change its resolution (see fig. 1). Assuming, as previously done, that the noise cancels the drift force exactly, the equations of motion reduce to t ¨ α = FN Kλ (t − t′ )vα (t′ )dt′ . (20) − mα x α −∞
Equation (20) describes a Hamiltonian scheme thermostatted with a GLE and, at the same time, a forcebased model F with a history-dependent friction term. We thus conclude that the removal of the drift force from the Hamiltonian force-field introduces a non-Markovian behavior whether the system is in thermal equilibrium, We now want to employ this information about the drift as in eq. (20), or not, eq. (18). Incidentally, we observe force fluctuations to eliminate them without disrupting a relation between the history-dependent friction term thermal equilibrium. Specifically, we observe that the and previously developed adaptive resolution simulation 30007-p4
A unified framework for force-based and energy-based adaptive resolution simulations methods [27]: in this case, a conceptually similar ad hoc term was introduced together with a local thermostat to enforce thermodynamical stability. To numerically validate eq. (20) we performed a simulation in which the GLE friction kernel Kλ (t − t′ ) has been approximated in terms of the drift force autocor dr (t)F
dr (0) reported in relation functions Cλ (t) = F fig. 1. The autocorrelations have been fitted with simple decaying exponentials; the decay time and the initial value Cλ (0) for intermediate values of λ have been obtained by interpolating the measured quantities with cubic splines. The numerical implementation of the GLE has been performed following the extended variable method as described in ref. [49]. Here we made use of the property of the system that the force autocorrelation functions decay before the molecules significantly change their position and therefore their resolution λ. In the integration, i.e. throughout the time during which the exponentially decaying kernel is non-zero, λ can in fact be considered constant. The results of this simulation are reported in fig. 2: specifically, in panel a) the energy of the system is shown as a function of the simulated time for the case of model F with GLE friction; for comparison, the same quantity has been measured in absence of the friction. In this second case the energy increases steadily with a rate of E˙ = 1.86 · 10−4 ǫ/τ per particle, which in the first case drops to E˙ = 3.88 · 10−5 ǫ/τ per particle; the nonperfect conservation of the energy can be attributed to the approximation that has been employed to fit and model the memory kernel. It is nonetheless worth noting that the energy increase at the end of this fairly long simulation is of the order of 2% for the model with GLE friction, which was obtained by approximating the kernel with no free parameters. For comparison we introduced in model F a standard friction term −γv acting only in the hybrid region. We then fine-tuned its value so to obtain an energy increase rate as close as possible to the one measured in the simulation with the GLE. This procedure led us to a friction γ = 5 · 10−5 m/τ , corresponding to a rate E˙ = 3.63·10−5 ǫ/τ per particle, not far from what was observed in the GLE friction case. The difference between the two methods lies in the fact that the GLE friction approach is based on a fundamental understanding of the physics of the system which is intrinsically non-Markovian; the simple friction approach, on the other hand, is just a oneparameter fit obtained by trial and error. It must be observed, though, that by all practical criteria the two lead to identical results: comparing the density profiles of the liquid in the two cases, fig. 2(b), we observe no relevant difference. In summary, we have demonstrated that the energybased H-AdResS scheme, thermostatted by means of a GLE, is formally equivalent to the force-based approach obtained from it by removal of the drift force
Fig. 2: (Color online) Panel (a): energy of the model F normalized to its initial value as a function of time, in the absence of any friction, with GLE friction, and with simple friction proportional to the instantaneous velocity. Panel (b): density profiles of the simulations with GLE friction and with simple friction. The average density has been computed over the last 4/5 of the simulations (the noisy profile being a consequence of little statistics). Note that the system is effectively out of equilibrium as the energy is not constant.
and complemented with a history-dependent friction. A numerical verification of the formalism has been carried out within the boundaries of an approximate fit for the memory kernel, which largely reduces the energy drift. Because of the short decay time of the friction kernel, and the weakness of a standard friction term providing equivalent results, the memory effects introduced by eliminating the drift force from a Hamiltonian scheme would be strongly suppressed by a conventional Langevin thermostat. This would be especially the case when the low-resolution model is parametrized to match the thermodynamic properties of the high-resolution model. This result justifies the use of the thermostatted force-based scheme obtained from the H-AdResS method, and possibly also of other, similar approaches. The interpretation of the drift force as a colored noise enables us to establish a formal, bottom-up connection between energy-based and force-based adaptive resolution simulation approaches. This unified framework provides novel insight in the theoretical foundation of a simulation paradigm whose main actors, force-based and energy-based approaches, are traditionally presented as alternative though complementary strategies. A deeper comprehension of the relation between them can thus provide a solid basis for better understood and more efficient computer simulation strategies.
30007-p5
Karsten Kreis et al. ∗∗∗ KK is recipient of a fellowship funded through the Excellence Initiative (DFG/GSC 266). The authors are indebted with R. Cortes Huerto and R. Everaers for a careful reading of the manuscript.
REFERENCES [1] Warshel A. and Levitt M., J. Mol. Biol., 103 (1976) 227. [2] Gao J., Lipkowitz K. and Boyd D., Methods and Applications of Combined Quantum Mechanical and Molecular Mechanical Potentials (Wiley) 1995. [3] Svensson M., Humbel S., Froese R., Matsubara T., Sieber S. and Morokuma K., J. Phys. Chem., 100 (1996) 19357. [4] Carloni P., Rothlisberger U. and Parrinello M., Acc. Chem. Res., 35 (2002) 455. [5] Kohlhoff S., Gumbsch P. and Fischmeister H. F., Philos. Mag., 64 (1991) 851. [6] Abraham F. F., Broughton J. Q., Bernstein N. and Kaxiras E., Comput. Phys., 12 (1998) 538. [7] Abraham F. F., Broughton J. Q., Bernstein N. and Kaxiras E., Europhys. Lett., 44 (1998) 783. [8] Broughton J. Q., Abraham F. F., Bernstein N. and Kaxiras E., Phys. Rev. B, 60 (1999) 2391. [9] Tadmor E. B., Phillips R. and Ortiz M., Langmuir, 12 (1996) 4529. [10] Shenoy V. B., Miller R., Tadmor E. B., Phillips R. and Ortiz M., Phys. Rev. Lett., 80 (1998) 742. [11] Shenoy V. B., Miller R., Tadmor E. B., Rodney D., Phillips R. and Ortiz M., J. Mech. Phys. Solids, 47 (1999) 611. [12] Rudd R. and Broughton J., Phys. Status Solidi B-Basic Res., 217 (2000) 251. [13] Rottler J., Barsky S. and Robbins M., Phys. Rev. Lett., 89 (2002) 148304. [14] Wagner G. J. and Liu W. K., J. Comput. Phys., 190 (2003) 249. [15] Xiao S. P. and Belytschko T., Comput. Methods Appl. Mech. Eng., 193 (2004) 1645. [16] Nie X., Chen S. and Robbins M. O., Phys. Fluids, 16 (2004) 3579. [17] Nie X., Chen S., E W. N. and Robbins M. O., J. Fluid Mech., 500 (2004) 55. [18] Nie X., Robbins M. O. and Chen S., Phys. Rev. Lett., 96 (2006) 134501. [19] Jiang D. and Carter E., Acta Mater., 52 (2004) 4801. [20] Lu G., Tadmor E. and Kaxiras E., Phys. Rev. B, 73 (2006) 024108. [21] Csanyi G., Albaret T., Payne M. C. and Vita A. D., Phys. Rev. Lett., 93 (2004) 175503. [22] Bernstein N., Varnai C., Solt I., Winfield S. A., Payne M. C., Simon I., Fuxreiter M. and Csanyi G., Phys. Chem. Chem. Phys., 14 (2012) 646.
[23] Bulo R. E., Ensing B., Sikkema J. and Visscher L., J. Chem. Theory Comput., 5 (2009) 2212. [24] Fleurat-Lessard P., Michel C. and Bulo R. E., J. Chem. Phys., 137 (2012). [25] Praprotnik M., Delle Site L. and Kremer K., J. Chem. Phys., 123 (2005) 224106. [26] Fritsch S., Poblete S., Junghans C., Ciccotti G., Delle Site L. and Kremer K., Phys. Rev. Lett., 108 (2012) 170602. [27] Ensing B., Nielsen S., Moore P., Klein M. and Parrinello M., J. Chem. Theory Comput., 3 (2007) 1100. [28] Poma A. B. and Site L. D., Phys. Rev. Lett., 104 (2010) 250201. [29] Potestio R. and Delle Site L., J. Chem. Phys., 136 (2012) 054101. [30] Mukherji D. and Kremer K., Macromolecules, 46 (2013) 9158. [31] Heyden A. and Truhlar D. G., J. Chem. Theory Comput., 4 (2008) 217. [32] Park J. H. and Heyden A., Mol. Simul., 35 (2009) 962. ˜ ol P., Delgado[33] Potestio R., Fritsch S., Espan Buscalioni R., Kremer K., Everaers R. and Donadio D., Phys. Rev. Lett., 110 (2013) 108301. ˜ ol P., Delgado-Buscalioni R., [34] Potestio R., Espan Everaers R., Kremer K. and Donadio D., Phys. Rev. Lett., 111 (2013) 060601. [35] Delgado-Buscalioni R., Kremer K. and Praprotnik M., J. Chem. Phys., 128 (2008) 114110. [36] Praprotnik M., Poblete S., Delle Site L. and Kremer K., Phys. Rev. Lett., 107 (2011) 099801. [37] Delle Site L., Phys. Rev. E, 76 (2007) 047701. ¨ tte C. and Delle Site [38] Wang H., Hartmann C., Schu L., Phys. Rev. X, 3 (2013) 011018. [39] Fritsch S., Junghans C. and Kremer K., J. Chem. Theory Comput., 8 (2012) 398. [40] Kirkwood J., J. Chem. Phys., 3 (1935) 300. [41] Weeks J. D., Chandler D. and Andersen H. C., J. Chem. Phys., 54 (1972) 5237. [42] Halverson J. D., Brandes T., Lenz O., Arnold A., Bevc S., Starchenko V., Kremer K., Stuehn T. and Reith D., Comput. Phys. Commun., 184 (2013) 1129. [43] Praprotnik M., Poblete S., Kremer K., J. Stat. Phys., 145 (2011) 946. [44] Matysiak S., Clementi C., Prapraotnik M., Kremer K., Delle Site L., J. Chem. Phys., 128 (2008) 024503. [45] Prapraotnik M., Matysiak S., Delle Site L., Kremer K., Clementi C., J. Phys.: Condens. Matter, 19 (2007) 292201. [46] Zwanzig R., J. Stat. Phys., 9 (1973) 215. [47] Illuminati F., Patriarca M. and Sodano P., Physica A: Stat. Mech. Appl., 211 (1994) 449. [48] Fox R. F., J. Math. Phys., 18 (1977) 2331. [49] Baczewski A. D. and Bond S. D., J. Chem. Phys., 139 (2013) 044107. [50] Krajewski F. R. and Parrinello M., Phys. Rev. B, 73 (2006) 041105. [51] Attaccalite C. and Sorella S., Phys. Rev. Lett., 100 (2008) 114501.
30007-p6
